Barbara Oakley, an engineering professor at Oakland University in Rochester, Mich., says the key to mathematics expertise is practice, not conceptual understanding as some common-core proponents would have educators believe.

She wrote last week in the Wall Street Journal that many teachers struggling with teaching the Common Core State Standards in math have been led to believe that mastery comes from an "ah-ha" moment of understanding. But she writes:

...[T]he development of true expertise involves extensive practice so that the fundamental neural architectures that underpin true expertise have time to grow and deepen. This involves plenty of repetition in a flexible variety of circumstances. In the hands of poor teachers, this repetition becomes rote—droning reiteration of easy material. With gifted teachers, however, this subtly shifting and expanding repetition mixed with new material becomes a form of deliberate practice and mastery learning.

In the common-core document, it says that students need "a solid understanding of concepts, a high degree of procedural skill and fluency, and the ability to apply the math." (This is collectively known as "rigor," one of the common standards' key shifts.) But the conceptual understanding piece seems to have gotten the most attention latelyin part because teaching this kind of higher-level thinking is (pretty much inarguably) more challenging than teaching a procedure.

Oakley goes on to write, "True mastery doesn't mean you use crutches like laying out 25 beans in 5-by-5 rows to demonstrate that 5 × 5 = 25. It means that when you see 5 × 5, in a flash, you know it's 25it's a single neural chunk that's as easy to pull up as a ribbon."

To be clear, the common core does not require that students use painstaking methods to calculate equationsin fact, 3rd graders are expected to "know from memory all products of two one-digit numbers" by the end of the year. But the standards do require that students understand a variety of visual representations for division and multiplication as well.

In any case, here's the kicker of Oakley's pieceand a solid argument for being mindful about weighing the emphasis on procedural fluency versus conceptual understanding in the math classroom: "Having students stop to continually check and prove their understanding can actually impede their understanding, in the same way that continually focusing on every aspect of a golf swing can impede the development of the swing."

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